leon.magiera at wp.pl writes:
> Problem 2
>
> ex:1/(sqrt((x-b)^2+a^2)*sqrt((x-b)^2/a^2+1));
>
> integrate(ex,x);
>
> Is the above integral too hard ?
For this, have a look at what your expression is. Firstly, the
x-b looks unpleasant so lets ignore that. Note that
integrate (f(x - b), x, u, v) = integrate (f(x), x, u-b, v-b)
and indeed you're asking for the indefinite integral, so this comes out
as something like
(integrate (f(x - b), x) at x=v) = (integrate (f(x), x) at x=v-b)
Anyway, that lets us get rid of the "-b" for a bit:
(%i1) ex:1/(sqrt((x-b)^2+a^2)*sqrt((x-b)^2/a^2+1))$
(%i2) no_b: subst(x, x-b, ex);
1
(%o2) --------------------------
2
2 2 x
sqrt(x + a ) sqrt(-- + 1)
2
a
Hmm. Now it looks like there should be some cancelling going on here. I
admit I did this on paper, but Maxima can do it too:
(%i3) %, ratsimp;
abs(a)
(%o3) -------
2 2
x + a
And that's easy enough to integrate (unless, like me, you've forgotten
the tables of integrals you learnt at A-level. Fortunately Maxima
hasn't):
(%i4) E_no_b: integrate (%, x);
x
abs(a) atan(-)
a
(%o4) --------------
a
Ahah! Now remember what I originally said about how b gets involved and
do the substitution by hand:
(%i5) E: subst(x-b, x, %);
x - b
abs(a) atan(-----)
a
(%o5) ------------------
a
Of course, abs(a)/a is ?1. If you know the sign of a, you can get even
further:
(%i6) assume (a>0);
(%o6) [a > 0]
(%i7) E, ev;
x - b
(%o7) atan(-----)
a
Tada! Now suppose you hadn't spotted the x-b thing. Fortunately, Maxima
can still cope: you just have to tell it to get rid of the square roots
by simplifying the product:
(%i8) forget (a>0);
(%o8) [a > 0]
(%i9) ratsimp(ex);
abs(a)
(%o9) --------------------
2 2 2
x - 2 b x + b + a
(%i10) integrate(%,x);
2 x - 2 b
abs(a) atan(---------)
2 a
(%o10) ----------------------
a
And you can get rid of the unfortunate factor of 2 with another call to
ratsimp if you fancy:
(%i11) ratsimp(%);
x - b
abs(a) atan(-----)
a
(%o11) ------------------
a
Rupert
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